%\clearpage\section{Broadcast with Network Coding}
%Network coding should outperform “naive” broadcast, we try to illustrate this. An expression for how many transmissions are neccesary to ensure that all nodes have received at least a number of packets with an error probability. It is assumed that all packets which are transmitted are linear independent (infinite field), thus ignoring the probability of receiving two or more linear dependent packets. The problem for one node can be solved by using the probability mass function, which is given in \eqref{eq:nc_1node}.

%\begin{align}
%P(k,n,p)&=\frac{n!}{(n-k)!k!}\cdot p^k\cdot (1-p)^{n-k}\label{eq:nc_1node}
%\intertext{Where:}
%n&\text{ is the total number of transmissions}\notag\\
%k&\text{ is the number of received packets from $n$ transmissions}\nonumber\\
%p&\text{ is the probability of successful transmissions}\notag
%\end{align}

%In \eqref{eq:nc_nnodes} the probability mass function is extended as to calculate $P_A$ as follows:

%\begin{align}
%P_A&=\left(\sum_{n=m}^{r}P(r,n,p(s))\right) ^j \label{eq:nc_nnodes}
%\intertext{Where:}
%P_A&\text{ is the probability that all nodes got at least $m$ packets}\notag\\
%m&\text{ is the minimum number of packets needed}\notag\\
%p(s)&\text{ is the probability of transmission success for a node}\notag\\
%j&\text{ is the number of nodes}\notag\\
%r&\text{ is the total number of transmitted packets}\notag
%\end{align}


%\subsection{Note on linear dependency and finite field}
%In the previous chapter infinite field was assumed. In \eqref{eq:lineardependency} the probability for linear dependency in a square matrix with size $g$ with randomly generated values from the field, $\mathbb{F}$, is given. \cite{NCMOBDEV_09}

%\begin{align}
%P&=1-\prod_{i=1}^{g}\left(1-\frac{1}{q^{i}}\right)\label{eq:lineardependency}
%\intertext{Where:}
%P&\text{ is the probability of less than full rank}\notag\\
%g&\text{ is the generation size}\notag\\
%q&\text{ is the field size, e.g 256 for $\mathbb{F}_{2^8}$}\notag
%\end{align}

%\clearpage
